Recently, multi-reference entropy model has been proposed, which captures channel-wise, local spatial, and global spatial correlations. Previous works adopt attention for global correlation capturing, however, the quadratic cpmplexity limits the potential of high-resolution image coding. In this paper, we propose the linear complexity global correlations capturing, via the decomposition of softmax operation. Based on it, we propose the MLIC$^{++}$, a learned image compression with linear complexity for multi-reference entropy modeling. Our MLIC$^{++}$ is more efficient and it reduces BD-rate by 12.44% on the Kodak dataset compared to VTM-17.0 when measured in PSNR. Code will be available at https://github.com/JiangWeibeta/MLIC.

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